where \(\widehat{^N\!G}^i_{\:kl}=G^i_{\:kl}-\delta ^i_{(k}\partial _{l)}\ln N\)  are components of a projectively equivalent connection which in general is not metric The metricity of such connection is related with Lie problem and bihamiltonian Stäckel systems, see e.g. [1]}, [2]}.. We recall that two metrics are called projectively equivalent if they have the same geodesics as un-parametrized curves.
